Space-variant subwavelength dielectric grating and applications thereof

ABSTRACT

An optical device includes a planar subwavelength grating ( 10 ) formed in a dielectric material and having a laterally varying, continuous grating vector. When used to modulate a beam of laterally uniform polarized electromagnetic radiation incident thereon, the device passes the incident beam with a predetermined, laterally varying transmissivity and/or retardation. When used to effect polarization state transformation, the device transforms a beam of electromagnetic radiation incident thereon into a transmitted beam having a predetermined, laterally varying polarization state. The device ( 214 ) can be used to provide radially polarized electromagnetic radiation for accelerating subatomic particles or for cutting a workpiece. The device ( 108 ) also can be used, in conjuction with a mechanism for measuring the lateral variation of the intensity of the transmitted beam, for measuring all four Stokes parameters that define the polarization state of the incident beam.

FIELD AND BACKGROUND OF THE INVENTION

[0001] The present invention relates to the production and manipulationof optically polarized light and, more particularly, to a dielectricgrating whose grating vector varies continuously laterally andapplications of this grating.

[0002] Laterally varying polarizers have found application in a varietyof fields, including optical communication, optical computers, materialprocessing, tight focusing, polarimetry, particle trapping and particleacceleration. For the most part, the transmission axes of thesepolarizers vary laterally in a discontinuous manner. For example, BahramJavidi and Takanori Nomura, “Polarization encoding for optical securitysystems”, Optical Engineering vol. 39 no. 9 pp. 2439-2443 (2000),perform polarization encoding using a polarization mask that consists ofa rectangular array of small linear polarizers, oriented randomly atangles between 0° and 180°. N. Davidson et al. “Realization of perfectshuffle and inverse perfect shuffle transforms with holographicelement”, Applied Optics vol. 31 no. 11 pp. 1810-1812 (1992), invert anoptical perfect shuffle using an interlaced polarizing mask that is aone-dimensional array of linear polarizers oriented alternately at 0°and 90°. Uwe D. Zeitner et al., “Polarization multiplexing ofdiffractive elements with metal-stripe grating pixels”. Applied Opticsvol. 38 no. 11 pp. 2177-2181 (1999), do optical encryption bypolarization multiplexing using an element array, some of whose elementsare linear polarizers oriented at 0° and 90°.

[0003] Gregory P. Nordin et al., “Micropolarizer array for infraredimaging polimetry”. Jornal of the Optical Society of America vol. 16 no.5 pp. 1168-1174 (1999) do polimetry using an array of micropolarizerswhose unit cell includes two 0° linear polarizers, one 90° linearpolarizer and one 1350 linear polarizer.

[0004] Franco Gori, “Measuring Stokes parameters by means of apolarization grating” Optics Letters vol. 24 no. 9 pp. 584-586 (1999)suggested using a polarizer whose transmission axis varies continuouslylaterally for the purpose of measuring the polarization state of a lightbeam. The embodiment of the polarizer actually suggested by Gori is onlystepwise continuous: adjacent parallel strips of linearly polarizingfilm, with each strip's transmission axis tilted relative to itsneighbors, so that the transmission axis of this polarizer is constantwithin each strip and discontinuous between strips.

[0005] Discontinuities in the lateral variation of the transmission axisof a polarizer can produce diffractions which degrade the opticalefficiency of the polarizer. There is thus a widely recognized need for,and it would be highly advantageous to have a polarizer whosetransmission axis varies laterally in a truly continuous manner.

[0006] Metal wire gratings long have been used as polarizers. When theperiod of a metal wire grating is much smaller than the incidentwavelength, the grating acts as a polarizer, reflecting all the lightpolarized parallel to the wires (TE mode) and transmitting only lightpolarized perpendicular to the wires (TM mode). For larger, but stillsubwavelength, periods, some TE mode light is transmitted, and it isnecessary to use vectorial solutions of Maxwell's equations to predictthe behavior of such gratings. This typically is done using RigorousCoupled Wave Analysis (RCWVA) (M. G. Moharam and T. K. Gaylord,“Rigorous coupled-vave analysis of metallic surface-relief gratings”,Journal of the Optical Society of America, part A vol. 3 pp. 1780-1787(1986).

SUMMARY OF THE INVENTION

[0007] According to the present invention there is provided an opticaldevice, for manipulating incident light of at most a certain maximumwavelength, including a substantially planar grating formed in adielectric material and having a space-variant, continuous gratingvector, at least a portion of the grating having a local period lessthan the maximum wavelength of the incident light.

[0008] According to the present invention there is provided a method ofmodulating laterally uniform, polarized light of at most a certainmaximum wavelength, including the steps of: (a) solving an equation□×{overscore (K)}(K₀,β)=0 for a grating vector {overscore (K)} that isdefined by a wavenumber K₀ and by a direction β relative to a referencedirection, the modulation depending on β, {overscore (K)} being suchthat at least a portion of a grating fabricated in accordance with{overscore (K)} has a local period less than the maximum wavelength ofthe light; (b) fabricating the grating in a dielectric material inaccordance with the grating vector {overscore (K)}; and (c) directingthe light at the grating.

[0009] According to the present invention there is provided a method ofimposing a laterally varying polarization state on light of at most acertain maximum wavelength, including the steps of: (a) solving anequation ∇×{overscore (K)}(K₀,β)=0 for a grating vector {overscore (K)}that is defined by a wavenumber K₀ and by a direction fi relative to areference direction, the laterally varying polarization state being atleast partially defined by p, {overscore (K)} being such that at least aportion of a grating fabricated in accordance with {overscore (K)} has alocal period less than the maximum wavelength of the light: (b)fabricating the grating in a dielectric material in accordance with{overscore (K)}; and (c) directing the light at the grating.

[0010] According to the present invention there is provided a method ofmeasuring a polarization state of light of at most a certain maximumwavelength, including the steps of: (a) providing a substantially planargrating having a transmission axis that varies in one lateral direction,at least a portion of the grating having a local period less than themaximum wavelength of the light; (b) directing the light at the grating;(c) measuring an intensity of the light that has traversed the grating;and (d) determining all four Stokes parameters of the light from theintensity.

[0011] According to the present invention there is provided a method ofmeasuring a polarization state of light of at most a certain maximumwavelength, including the steps of: (a) providing a substantially planargrating having a reflection axis that varies in one lateral direction,at least a portion of the grating having a local period less than themaximum wavelength of the light; (b) directing the light at the grating:(c) measuring an intensity of the light that is reflected from thegrating; and (d) determining all four Stokes parameters of the lightfrom the intensity.

[0012] According to the present invention there is provided an opticaldevice, for transforming an incident beam of light into a transformedbeam of light, including a substantially planar grating formed in adielectric material and having a space-variant continuous gratingvector, such that the transformed beam is substantially free ofpropagating orders higher than zero order.

[0013] According to the present invention there is provided a method oftransforming an incident beam of laterally uniform, polarized light intoa modulated transmitted beam, including the steps of: (a) solving anequation ∇×{overscore (K)}(K₀, β)=0 for a grating vector {overscore (K)}that is defined by a wavenumber K₀ and by a direction 8 relative to areference direction, the modulation depending on β, {overscore (K)}being such that the transmitted beam is substantially free ofpropagating orders higher than zero order; (b) fabricating the gratingin a dielectric material in accordance with the grating vector{overscore (K)}; and (c) directing the light at the grating.

[0014] According to the present invention there is provided a method oftransforming an incident light beam into a transmitted beam upon whichis imposed a laterally varying polarization state, including the stepsof: (a) solving an equation ∇×{overscore (K)}(K₀,β)=0 for a gratingvector {overscore (K)} that is defined by a wavenumber K₀ and by adirection 6 relative to a reference direction, the laterally varyingpolarization state being at least partially defined by β, {overscore(K)} being such that the transmitted beam is substantially free ofpropagating orders higher than zero order; (b) fabricating the gratingin a dielectric material in accordance with {overscore (K)}; and (c)directing the light at the grating.

[0015] According to the present invention there is provided a method ofmeasuring a polarization state of an incident light beam, including thesteps of: (a) providing a substantially planar grating having atransmission axis that varies in one lateral direction, the gratingbeing operative to transform the incident beam into a transmitted beamthat is substantially free of propagating orders higher than zero order;(b) directing the light at the grating; (c) measuring an intensity ofthe transmitted beam; and (d) determining all four Stokes parameters ofthe light from the intensity.

[0016] According to the present invention there is provided a method ofmeasuring a polarization state of an incident light beam, including thesteps of: (a) providing a substantially planar grating having areflection axis that varies in one lateral direction, the grating beingoperative to transform the incident beam into a reflected beam that issubstantially free of propagating orders higher than zero order; (b)directing the light at the grating; (c) measuring an intensity of thereflected beam; and (d) determining all four Stokes parameters of thelight from the intensity.

[0017] All references to “light” herein are to be understood asreferring to electromagnetic radiation generally, even though theprimary application of the present invention is to infrared light.

[0018] The optical device of the present invention is a planar grating,formed in a dielectric material, whose grating vector {overscore (K)}has a vanishing curl. The vector {overscore (K)} is defined by itsmagnitude: a wavenumber, or spatial frequency, K₀; and by its direction6 relative to a reference direction. In fact, β is the localtransmission axis of the grating. Either K₀ or β or both may varylaterally and continuously. A grating vector with the property thateither K₀ or β or both may vary laterally is denoted herein as a“space-variant” grating vector. The lateral variation may be periodic,for example, translationally periodic or rotationally periodic.

[0019] According to the present invention, the grating is asubwavelength grating, meaning that the local period of at least aportion of the grating is less than the maximum wavelength of the lightthat is manipulated by the grating. It is to be understood that thepresent invention may be used to manipulate either monochromatic lightor polychromatic light. All references herein to a “maximum” wavelengthare to the wavelength that is used to define the “subwavelength”natureof the grating (the local period of the grating is shorter than themaximum wavelength) and the “dielectric” nature of the material in whichthe grating is formed (the material is substantially transparent tolight of the maximum wavelength). One consequence of the subwavelengthnature of the grating is that the transmitted beam is substantially freeof propagating orders higher than zero order. The grating behaves as alayer of uniaxial crystal, with the optical axes of the crystalperpendicular and parallel to the grating grooves (S. Y. Chou and W.Dong. Applied Physics Letters vol. 67 pp. 742-744 (1995)).

[0020] Preferably, the grating is formed as a plurality of grooves in aplanar substrate of the dielectric material. Preferable dielectricmaterials include gallium arsenide and zinc selenide for infraredapplications, and quartz and silica glass for visible lightapplications.

[0021] Although the description herein is directed at transmissiongratings, it is to be understood that the scope of the present inventionincludes both transmission gratings and reflection gratings. Inparticular, the subwavelength nature of a reflection grating of thepresent invention leads to the reflected beam being substantially freeof propagating orders higher than zero order. It will be obvious to oneskilled in the art how to modify the teachings herein for the case ofreflection gratings.

[0022] When used to modulate the intensity and/or retardation oflaterally uniform polarized incident light, the optical device of thepresent invention is operative to pass or reflect the incident lightwith a predetermined, laterally varying transmissivity, reflectivity orretardation. Preferably, this transmissivity, reflectivity orretardation varies periodically in one lateral direction.

[0023] When used to effect polarization state transformations, theoptical device of the present invention is operative to transform lightincident thereon into a transmitted or reflected beam having apredetermined, laterally varying polarization state. Preferably, thetransmitted or reflected beam has an azimuthal angle that varieslinearly in one lateral direction. Alternatively, the transmitted orreflected beam is either radially polarized or azimuthally polarizedwith the radial or azimuthal polarization being either in-phase oranti-phase.

[0024] The scope of the present invention also includes a particleaccelerator that is based on the optical device of the presentinvention. Specifically, this particle accelerator includes: (a) asource of light; (b) a first optical mechanism for forming the lightinto an annular beam; (c) the optical device of the present invention,for imposing radial polarization on the annular beam; (d) a secondoptical mechanism for focusing the radially polarized annular beam ontoa focal region; and (e) a particle source for directing a beam of theparticles longitudinally through the focal region.

[0025] The scope of the present invention also includes a method ofcutting a workpiece. The optical device of the present invention is usedto impose radial polarization on a beam of light. The radially polarizedbeam is directed at the workpiece to cut the workpiece.

[0026] The scope of the present invention also includes an apparatus,for measuring the polarization state of light, that is based on theoptical device of the present invention. Specifically, this apparatusincludes, in addition to the optical device of the present invention, amechanism for measuring the lateral variation of the intensity of thelight after the light has been manipulated by the device of the presentinvention.

[0027] The scope of the present invention also includes a method ofmodulating the intensity and/or the retardation of laterally uniform,polarized light. The equation ∇×{overscore (K)}=0 is solved for thegrating vector {overscore (K)} whose direction β gives the desiredmodulation. A grating is fabricated in a dielectric material inaccordance with this grating vector, and the electromagnetic radiationis directed at the grating. Preferable dielectric materials includegallium arsenide and zinc selenide for infrared applications, and quartzand silica glass for visible light applications.

[0028] The scope of the present invention also includes a method ofimposing a laterally varying polarization state on light. The equation∇×{overscore (K)}=0 is solved for the grating vector {overscore (K)}whose direction β, either by itself or in conjunction with otherparameters, defines the laterally varying polarization state. A gratingis fabricated in a dielectric material in accordance with this gratingvector, and the electromagnetic radiation is directed at the grating.Preferable dielectric materials include gallium arsenide and zincselenide for infrared applications and quartz and silica glass forvisible light applications.

[0029] Preferably the reference direction for β is the radial directionof a polar (r, θ) coordinate system.

[0030] The scope of the present invention also includes a method ofmeasuring the polarization state of light. The light is directed at agrating that has a transmission or reflection axis that varies in onelateral direction. Although a piecewise continuous, laterally varyinggrating, such as Gori's grating, may be used, it is preferable to use agrating whose transmission or reflection axis varies continuously in theone lateral direction. Most preferably, the transmission or reflectionaxis of the grating varies linearly in the one lateral direction. Theintensity of the light that has traversed the grating is measured, andall four Stokes parameters of the light are determined from the measuredintensity, preferably by performing respective integral transforms ofthe measured intensity.

[0031] Preferably, at least a portion of the light that emerges from thegrating is caused to traverse a polarizer before the intensity of thelight is measured.

[0032] As noted above, the grating need not be an optical device of thepresent invention, although it is preferable that the grating besubstantially planar, be formed in a dielectric material, and have aspace-variant, continuous grating vector {overscore (K)} such that thetransmission or reflection axis is the direction β of {overscore (K)}.

BRIEF DESCRIPTION OF THE DRAWINGS

[0033] The invention is herein described, by way of example only, withreference to the accompanying drawings, wherein:

[0034]FIG. 1 illustrates the definition of the grating vector of asubwavelength grating;

[0035]FIGS. 2A, 2B and 2C are SEM images of a chirped grating;

[0036]FIG. 3A is a plot of the transmission coefficient of the chirpedgrating as a function of period;

[0037]FIG. 3B is a plot of the retardation of the chirped grating as afunction of period:

[0038]FIG. 5A is a schematic depiction of the geometry of a radialgrating;

[0039]FIG. 5B is an SEM image of a radial grating;

[0040]FIG. 6A shows the local azimuthal angles of a right circularlypolarized light beam following transmission through the grating of FIG.5B;

[0041]FIG. 6B shows the local azimuthal angles of a left circularlypolarized light beam following transmission through the grating of FIG.5B;

[0042]FIG. 7 is a schematic diagram of an apparatus of the presentinvention for measuring the polarization state of an incident lightbeam:

[0043]FIG. 8A is a plot of intensity, as a function of lateral positionin the apparatus of

[0044]FIG. 7, of an initially circularly polarized light beam thattransits a quarter wave plate;

[0045]FIG. 8B is a plot of the Stokes parameters of the light beam ofFIG. 8A, as measured by the apparatus of FIG. 7, as functions of theorientation of the quarter wave plate;

[0046]FIG. 8C depicts the Stokes parameters of FIG. 8B on a Poincaresphere;

[0047]FIG. 9A is a plot of the Stokes parameters, as measured by theapparatus of FIG. 7, of an initially right circularly polarized lightbeam that transits a rotating half wave plate, as functions of thecumulative rotation of the half wave plate;

[0048]FIG. 9B is a plot of the degree of polarization of the light beamof FIG. 9A as a function of the cumulative rotation of the half waveplate; and

[0049]FIG. 10 is a schematic diagram of an inverse Cerenkov acceleratorof the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0050] The present invention is of a subwavelength dielectric gratingwhich can be used to impose, on incident electromagnetic radiation, alaterally varying modulation of intensity and/or retardation, or alaterally varying polarization state.

[0051] The principles and operation of a subwavelength dielectricgrating according to the present invention may be better understood withreference to the drawings and the accompanying description.

[0052] Gori defined his “polarization grating” as “a transparency inwhich the polarization of the incident wave is changed periodicallyalong a line”. The present invention is somewhat more general: anoptical device, fabricated in a dielectric material, that imposes a (notnecessarily periodic) laterally varying modulation of intensity and/orretardation, or alternatively a (not necessarily periodic) laterallyvarying polarization state, on an incident beam of electromagneticradiation that is laterally uniform in intensity and polarization state.The lateral variation may be in one or both orthogonal directionstransverse to the direction of propagation. As will be seen, lateralvariation in only one transverse direction is an important special case.Alternatively, the lateral variation may be along the radial and/orazimuthal directions of a circular coordinate system.

[0053] Referring now to the drawings. FIG. 1 illustrates the definitionof the grating vector {overscore (K)}of a subwavelength grating 10 onwhich is incident a beam of electromagnetic radiation. Grating 10consists of (locally) parallel grooves 14 in a planar dielectricsubstrate. The plane of the substrate is parallel to the x,y plane of aCartesian (x,y,z) coordinate system. (See FIGS. 2A, 2B, 2C and 5B belowfor Scanning Electron Microscope (SEM) images of examples of suchgrooves in planar substrates.) For clarity, only four grooves 14 areshown in FIG. 1. The period of grating 10 is Λ. The direction ofpropagation of the electromagnetic radiation is perpendicular to grating10, i.e., in the z-direction. The grating vector {overscore (K)} isdefined by its magnitude K₀=2π/Λ and by its direction β relative to the+x-axis as a reference direction.

[0054] As noted above, the gratings of the present invention aresubwavelength gratings, i.e., gratings 10 whose local period Λ is lessthan the wavelength of the incident beam.

[0055] Also as noted above, when the period of a dielectric grating issmaller than the incident wavelength, only the zeroth order is apropagating order, and the grating behaves as a layer of uniaxialcrystal, with the optical axes perpendicular and parallel to the gratinggrooves. Therefore, by controlling the local period, structure anddirection of the grating, any desired space-variant waveplate can becreated.

[0056] Understanding of the present invention is facilitated byconsidering one of the simplest embodiments thereof: a chirp grating,i.e., a grating in which β is constant and Λ varies linearly in thex-direction: Λ=Λ₀+bx. The grating vector {overscore (K)} then is:$\begin{matrix}{\overset{\rightharpoonup}{K} = {\frac{2\quad \pi}{\Lambda_{0} + {bx}}\hat{x}}} & (1)\end{matrix}$

[0057] where {circumflex over (x)} is a unit vector in the +x-direction.

[0058] A Lee-type (W. H. Lee, “Binary synthetic holograms”, AppliedOptics vol. 13 pp. 1677-1682 (1974)) binary chirped grating was realizedas a 5 millimeter by 3 millimeter rectangle with a local period Avarying from 2 microns to 10 microns. A chrome mask of the grating wasfabricated using high-resolution laser lithography. The pattern wastransferred to a 500 micron thick GaAs wafer using photolithography.Then the grating was etched using electron cyclotron resonance with BC1₃for 35 minutes, to a depth of about 2.5 microns, in order to obtain aretardation of 10.6 micron radiation of approximately π/2 at shortperiods Λ. Finally, an anti-reflection coating was applied to the backof the wafer. FIGS. 2A, 2B and 2C are SENM images of the chirped gratingat periods Λ of 2 microns, 3 microns and 8 microns, respectively.

[0059] To determine the dependence of the transmission and retardationof the chirped grating on the local period Λ, the grating wasilluminated with light from a CO₂ laser at a free space wavelength of10.6 microns and the dependence of the transmission coefficients t_(x)and t_(y) and of the retardation φ on the grating period Λ were measuredusing ellipsometric techniques (E. Collet, Polarized Light (MarcelDekker, New York, 1993)). FIG. 3A shows the measured transmissioncoefficients of the chirped grating along with the theoretical resultscalculated using RCWA. There is good agreement between the calculationsand the measurements. FIG. 3A shows that when the grating period A isbetween 2 microns and 3.24 microns, t_(x) and t_(y) do not vary much andhave values of around 0.95 and 0.82 respectively. At a period Λ of 3.24microns (10.6 microns divided by the index of refraction of GaAs), thereis anomalous behavior due to non-zero propagating orders within thewafer, and the transmission drops sharply. FIG. 3B shows the measuredand calculated retardation φ of the chirped grating at all gratingperiods Λ. φ is close to π/2 at grating periods A between 2 microns and4 microns. At longer periods, φ begins to vary.

[0060] The results of FIGS. 2 and 3 now will be applied to the design ofa space-variant waveplate whose transmission axis that varies linearlyalong the x-direction and that is described by the grating vector

{overscore (K)}(x, y)=K ₀(x, y)cos(ax){circumflex over (x)}+K ₀(x,y)sin(ax)ŷ  (2)

[0061] where {circumflex over (x)} is a unit vector in the +x-direction,as before, and ŷ is a unit vector in the +y direction. In order for thisgrating to be physically realizable, the grating vector must have avanishing curl, so that $\begin{matrix}{{\frac{\partial K_{0}}{\partial y}{\cos ({ax})}} = {{\frac{\partial K_{0}}{\partial x}{\sin \left( {a\quad x} \right)}} + {\beta \quad K_{0}\quad {\cos ({ax})}}}} & (3)\end{matrix}$

[0062] Equation (3) can be solved by equating the coefficients ofcos(ax) and sin(ax) to zero independently, resulting in the gratingvector $\begin{matrix}{\overset{\rightharpoonup}{K} = {\frac{2\quad \pi}{\Lambda_{0}}{{\exp \left( {a\quad y} \right)}\left\lbrack {{{\cos ({ax})}\hat{x}} + {{\sin ({ax})}\hat{y}}} \right\rbrack}}} & (4)\end{matrix}$

[0063] where Λ₀ is the period at ∂=0. The corresponding grating functionφ(x,y), whose gradient is the grating vector, is found by integratingthe grating vector along an arbitrary path in the x,y plane:$\begin{matrix}{{\varphi \left( {x,y} \right)} = {\frac{2\quad \pi}{a\quad \Lambda_{0}}{\sin ({ax})}\quad {\exp ({ay})}}} & (5)\end{matrix}$

[0064] Equations (4) and (5) show that the constraint on the continuityof the grating results in a varying period, which depends on they-coordinate. Therefore, the retardation (and the transmissioncoefficients t_(x) and t_(y), which depend on the period, also vary inthe y-direction.

[0065] The design of a subwavelength dielectric grating for imposing aradially or azimuthally varying polarization state now will bediscussed. By correctly determining the direction, period and depth ofthe grating, any desired continuous polarization can be obtained.Furthermore the continuity of the grating ensures the continuity of thetransmitted field, thus suppressing diffraction effects that arise fromdiscontinuity.

[0066] To obtain a radially or azimuthally varying polarization state,the grating vector must be expressed in circular (r,θ) coordinates:

{overscore (K)}=K ₀(r,θ) cos(β(r,θ)){circumflex over(r)}+K₀(r,θ)sin(β(r,θ)){overscore (θ)}  (6)

[0067] where {circumflex over (r)} is a unit vector in the radialdirection and θ is a unit vector in the azimuthal direction. Thedirection β of the grating vector now is relative to the local radialdirection as a reference direction.

[0068] FIGS. 4A-4D illustrates the four kinds of polarization states ofinterest: in-phase radial (FIG. 4A), anti-phase radial (FIG. 4B),in-phase azimuthal (FIG. 4C) and anti-phase azimuthal (FIG. 4D), withcontinuous electromagnetic fields. In FIGS. 4A and 4C, the fields atopposite sides of the center are in-phase and at any given instance theelectric fields at those points are of equal magnitude and are orientedin the same direction. This is as opposed to the fields in FIGS. 4B and4D, for which the electric fields at opposite sides of the circle areanti-phase, so that at any given instance, these fields possess the samemagnitude and are oriented in opposite directions. Because of thesymmetry of the beams, it is clear that the dark center of theanti-phase polarization is conserved during propagation, as opposed tothe in-phase polarization, which displays a bright center in the farfield. Both types of polarization can be produced by gratings of thepresent invention. If the incident beam is circularly polarized, thenthe grating should be followed by a spiral phase element to getanti-phase polarization.

[0069] The design of a “radial” grating for converting right handcircularly polarized light into radially polarized light now will bediscussed. This can be achieved using a space varying quarter waveplate,i.e., the depth and structure of the local grating is such that theretardation is π/2. In addition, β must be −45° at all points to ensurethat the resulting polarization is linearly polarized in the desireddirection. (Note that incident left hand circular polarized light isconverted to azimuthally polarized light.) Setting β equal to 45° inequation (6) and recalling that continuity demands that the curl of thegrating vector must vanish gives the following differential equation:$\begin{matrix}{{\frac{1}{r}\left( {{\frac{\partial}{\partial r}\left\lbrack {- {{rK}_{0}\left( {r,\theta} \right)}} \right\rbrack} + \frac{\partial{K_{0}\left( {r,\theta} \right)}}{\partial\theta}} \right)} = 0} & (7)\end{matrix}$

[0070] Because of the symmetry of the problem, it can be assumed that K₀is independent of θ a in which case equation (7) can be solved to vieldK₀(r)=(2πr₀/Λ₀)/r, where Λ₀ is the period when r=r₀. Integrating theresulting vector along an arbitrary path yields the grating function

φ=({square root}{square root over (2)}πr ₀/Λ₀)[ln(r/r ₀)−θ]   (8)

[0071] Continuity of this function requires that φ(r,θ)=φ(r,φ+2π)±2πm,where Λ is an integer, and therefore {square root}{square root over(2)}π r₀/Λ₀ must be an integer. This places a constraint on r₀ and Λ₀.

[0072] A Lee-type binary grating corresponding to the grating functionof equation (8) was realized as described above for the chirped grating,except that the duration of the etching, was 39 minutes. Λ₀ was 2microns and r₀ was 5 millimeters. The range of r was from 5 millimetersto 8 millimeters, and the range of Λ was from 2 microns to 3.2 microns,so as not to exceed the Wood anomaly. FIG. 5A is a schematic depictionof the geometry of this grating. This depiction is only schematic, inthe sense that the spacing of grooves 22 as shown in FIG. 5A is muchwider than any of the local periods of the actual grating. (With r₀=5millimeters and Λ₀=2 microns, there should be on the order of 15,000grooves 22 shown in FIG. 5A, rather than the 45 grooves 22 actuallyshown.) FIG. 5B shows a typical SEM cross section of the gratingprofile.

[0073] The grating corresponding to equation (11) was illuminated with10.6 micron right hand circularly polarized light from a CO₂ laser. Thetransmitted light was imaged onto a Spiricon Pyrocam I camera, and thefour Stokes parameters S₀, S₁, S₂ and S₃ were measured at each pointusing the four measurement technique (Collet, op. cit.). The localazimuthal angle ψ was calculated from tan(2ψ)=S₂,S₁. The localellipticity tanv was calculated from sin(2χ)=S₃/S₀. For right handcircularly polarized light, the average ellipticity was 0.08, and theaverage deviation of the azimuthal angle was 2.6°, yielding apolarization purity (percentage of energy in the desired polarization)of 99.2%. The transmission was 86%. FIG. 6A shows the local azimuthalangle of the resulting beam. The arrows in FIG. 6A show radialpolarization. FIG. 6B shows the local azimuthal angle when left handcircularly polarized incident light was used. The arrows in FIG. 6B showazimuthal polarization.

[0074] Three applications of the polarization grating of the presentinvention now will be discussed.

[0075] The first application is to polarimetry, i.e., the measurement ofthe polarization state of a light beam. Such measurements are used for alarge range of applications, including ellipsometry (A. N. Naciri etal., “Spectroscopic generalized ellipsometry based on Fourier analysis”,Applied Optics vol. 38 pp. 4802-4811 (1999)), biosensing (V. Sankaran etal. “Comparison of polarized light propagation in biological tissues andphantoms. Optics Letters vol. 24, pp. 1044-1046 (1999)), quantumcomputing (M. Koashi et al. “Probabilistic manipulation of entangledphotons”, Physical Review A vol. 63 article no. 0.3031 (2001)) andoptical communications (P. C. Chou et al. “Real time principal statecharacterization for use in PMD compensators”, IEEE Photon TechnologyLetters vol. 13 pp. 568-570 (2001)). One commonly used method ofpolarimetry is to measure the time dependent signal when the beam istransmitted through a rotating polarizer or quarter wave plate (Collet,pp. 103-107). By Fourier analysis of this signal, the Stokes parametersof the beam can be derived. This method is relatively slow, because itrelies on a series of consecutive measurements. This makes this methoddifficult to use in applications, such as polarization mode dispersioncompensation in optical communications, that require the measurement ofdynamic changes in polarization.

[0076] The waveplate whose grating vector is described by equation (5)can be used to implement the space domain analog of the rotatingpolarizer method. By performing a Fourier analysis of the transmittedintensity, the polarization state of the incident beam can be determinedin real time.

[0077]FIG. 7 is a schematic diagram of an apparatus 100 of the presentinvention for real time polarization measurements. A polarized lightbeam 102 from a laser 104 is incident on a polarization-sensitive medium106. Examples of such polarization-sensitive media include biologicaltissue, optical fibers, ellipsometric samples and waveplates. Beam 102then is transmitted through a subwavelength dielectric grating 108 ofthe present invention that is designed in accordance with equations (4)and (5), followed by a subwavelength metal wire polarizer 110. Theresulting space variant intensity distribution is imaged onto a camera112, and a Fourier analysis of the recorded space-variant intensity isperformed by a processor 114, yielding the polarization of beam 102incident on grating 108, thereby enabling analysis of the sample.

[0078] The polarization state of light can be described as a Stokesvector (S₀,S₁,S₂,S₃)_(T). In general, S₀ ²≧S₁ ²+S₂ ²+S₃ ², with equalityholding, only for a fully polarized beam. Waveplates and polarizers aredescribed in the Stokes representation by 4×4 Mueller matrices. Forexample, a waveplate with retardation (p and with real transmissioncoefficients tx, ty for its two eigen-polarizations is written as:$\begin{matrix}{W = {\frac{1}{2}\begin{bmatrix}{t_{x}^{2} + t_{y}^{2}} & {t_{x}^{2} - t_{y}^{2}} & 0 & 0 \\{t_{x}^{2} - t_{y}^{2}} & {t_{x}^{2} + t_{y}^{2}} & 0 & 0 \\0 & 0 & {2t_{x}t_{y}\quad \cos \quad \phi} & {{- 2}\quad t_{x}t_{y}\quad \sin \quad \phi} \\\quad & \quad & {2t_{x}t_{y}\quad \sin \quad \phi} & {2\quad t_{x}t_{y}\quad \cos \quad \phi}\end{bmatrix}}} & (9)\end{matrix}$

[0079] Consequently, grating 108 whose fast axis orientation varieslinearly in the x direction, followed by polarizer 110, can be describedby the space-varying Mueller matrix

M(x)=PR(−ax)WR(ax)  (10)

[0080] where a is the rate of rotation of the fast axis, $\begin{matrix}{{R({ax})} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad 2\quad {ax}} & {\sin \quad 2\quad {ax}} & 0 \\0 & {{- \sin}\quad 2\quad {ax}} & {\cos \quad 2\quad {ax}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (11)\end{matrix}$

[0081] is the Mueller matrix for a rotator, and $\begin{matrix}{P = {\frac{1}{2}\begin{bmatrix}1 & 1 & 0 & 0 \\1 & 1 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}}} & (12)\end{matrix}$

[0082] is the Mueller matrix for a polarizer.

[0083] Suppose that a monochromatic plane wave in an arbitrary state ofpolarization (S₀,S₁,S₂,S₃)^(T) is incident on the apparatus described byequation (10). Then the polarization state of the transmitted beam isspace varying, with a Stokes vector

(S₀′(x),S₁′(x),S ₂′(x),S ₃′(x))^(T) =M(x)(S₀,

[0084] The transmitted intensity, as a function of x, is $\begin{matrix}\begin{matrix}{{S_{0}^{\prime}(x)} = \left\{ {{A\quad S_{0}} + {\left( {A + C} \right){S_{1}/2}} + {{B\left( {S_{1} + S_{0}} \right)}\cos \quad 2\quad {ax}} +} \right.} \\{{{\left( {{BS}_{2} - {DS}_{3}} \right)\sin \quad 2\quad {ax}} +}} \\{\left. {\left( {A - C} \right){\left( {{S_{1}\cos \quad 4\quad {ax}} + {S_{2}\sin \quad 4\quad {ax}}} \right)/2}} \right\}/4}\end{matrix} & (14)\end{matrix}$

[0085] where A=t_(x) ²+t_(y) ², B t_(x) ²−t_(y) ², C=2t_(x)t_(y) cos φand D=2t_(x)t_(y) sin φ. Equation (14) describes a truncated Fourierseries whose coefficients depend on the Stokes parameters of theincident beam. Fourier analysis yields: $\begin{matrix}{{{A\quad S_{0}} + {\left( {A + C} \right){S_{1}/2}}} = {\frac{2a}{\pi}{\int_{0}^{2\quad {\pi/a}}{{S_{0}^{\prime}(x)}{x}}}}} & (15) \\{{B\left( {S_{1} + S_{0}} \right)} = {\frac{4\quad a}{\pi}{\int_{0}^{2\quad {\pi/a}}{{S_{0}^{\prime}(x)}{\cos \left( {2\quad {ax}} \right)}\quad {x}}}}} & (16) \\{{{B\quad S_{2}} - {D\quad S_{3}}} = {\frac{4a}{\pi}{\int_{0}^{2\quad {\pi/a}}{{S_{0}^{\prime}(x)}\sin \quad \left( {2\quad {ax}} \right)\quad {x}}}}} & (17) \\{{\left( {A - C} \right){S_{1}/2}} = {\frac{4a}{\pi}{\int_{0}^{2\quad {\pi/a}}{{S_{0}^{\prime}(x)}\cos \quad \left( {4\quad {ax}} \right)\quad {x}}}}} & (18) \\{{\left( {A - C} \right){S_{2}/2}} = {\frac{4a}{\pi}{\int_{0}^{2\quad {\pi/a}}{{S_{0}^{\prime}(x)}\quad {\sin \left( {4\quad {ax}} \right)}\quad {x}}}}} & (19)\end{matrix}$

[0086] from which all four Stokes parameters S₀, S₁,S₂ and S₃ can becalculated.

[0087] Equation (5) shows that the local period of grating 108 isindependent of x and increases exponentially with y. Based on thediscussion above of the chirped grating of the present invention, ifbeam 102 has a free space wavelength of 10.6 microns, then the portionof grating 108 that is actually used for polimetry should be limited tothe portion of grating 108 in which the local period is between 2microns and 3.24 microns. Because the local period of grating 108depends on y, the intensity measurements should be made along lines thatare parallel to the x-axis of grating 108, to preserve the validity ofequations (14) through (19). Preferably, the varying period of grating108 is used to reduce statistical measurement errors by performingseveral measurements, along several lines parallel to the x-axis ofgrating 108. Most preferably, these measurements are performedsimultaneously.

[0088] Apparatus 100 was realized using a CO₂ laser 104 that emittedlinearly polarized light at a free space wavelength of 10.6 microns andusing a grating 108, having a transmission axis that varied periodicallyin the x direction, fabricated as described above for the chirpedgrating. To test apparatus 100, a quarter wave plate (QWP) was used formedium 106. Camera 112 was a Spiricon Pyrocam I. FIG. 8A shows themeasured intensity distributions S₀′(x) at a period of 2.5 microns, aswell as the predicted results calculated from equation (17) when thefast axis of the QWP was at angles of −15°, 5° and 122.5°. The differentincident polarizations produce distinct intensity distributions fromwhich the corresponding Stokes parameters can be measured. FIG. 8B showsthe measured and predicted Stokes parameters of the resulting beam, as afunction of the orientation of the QWP. The experimental values weredetermined by fitting the curve of equation (17) to the measuredintensity distributions with S₁, S₂ and S₃ as free parameters. Themeasurements yielded an average error of 1° in the measured azimuthalangle R and an average error of 0.025 in the measured ellipticity tanχ.FIG. 8C depicts the same polarization measurements on a Poincare sphere.FIG. 8 provides a graphic representation that shows all four Stokesparameters simultaneously, thereby emphasizing the good agreementbetween the predictions and the measurements for polarized light.

[0089] Some applications, however, require analysis of partiallypolarized light. Such beams are characterized by their degree ofpolarization (DOP), defined as:

DOP={[<S ₁(t)>² +<S ₂(t)>² +<S ₃(t)>² ]/<S ₀(t)>²}^(1/2)  (23)

[0090] where t is time and “<>” denotes time-domain averaging. Iflinearly polarized light is incident on a half waveplate (HWP) rotatingat angular frequency ω, then S₁(t)=cos(4ωt), S₂(t)=sin(4ωt) and S₃(t)=0.The corresponding DOP is |(sin2Ω)/2Ω|, where Ω is the angle by which theHWP was rotated during the time span over which the averaging wasperformed.

[0091] To test the ability of apparatus 100 to analyze partialpolarization, a rotating HWP was used for medium 106 and ω was chosen sothat camera 112 captured an image each time the HWP rotated another2.5°. The average intensity was calculated at each pixel of camera 112,using all images captured during the rotation of the HWP up to an angleΩ. The appropriate Stokes parameters then were calculated. Based on theStokes parameters, the dependence of the DOP on Ω was calculated. FIG.9A shows the measured and predicted Stokes parameters as a function ofΩ. S3 is close to zero for all values of Ω. S1 and S2 tend to zero as Ωincreases. FIG. 9B shows the measured and predicted DOP. As Ω increases,both the measured DOP and the predicted DOP tend to zero.

[0092] The second application is to the acceleration of subatomicparticles. FIG. 10 is a schematic diagram of an inverse Cerenkovaccelerator 200 of the present invention. Accelerator 200 is similar tothe accelerators described by Y. Liu et al. in “Vacuum laseracceleration using a radially polarized CO₂ laser beam”. NuclearInstruments and Methods in Physics Research A vol. 424 pp. 296-303(1999) and by I. V. Pogorelsky et al. in “CO₂ laser technology foradvanced particle accelerators”, a web page whose URL is

[0093]http://nslsweb.nsls.bnl.gov/AccTest/publications/co2_laser_tech.htm.

[0094] A source 202 emits a beam 204 of electrons that are to beaccelerated. A carbon dioxide laser 206 emits a coherent beam 208 of10.6 micron light. Beam 208 is converted into an annular light beam 210by a negative axicon lens 216 and a positive axicon lens 218. An annularmirror 212 is placed to reflect annular light beam 210 parallel toelectron beam 204, with electron beam 204 traversing an aperture 224 inmirror 212 so that electron beam 204 travels along the axis of annularlight beam 210. A subwavelength dielectric grating 214 of the presentinvention, similar to the radial grating described above, followed by aspiral phase element 232 with a phase function exp[−iθ(xy)] (R. Oron etal., “Continuous phase elements can improve laser beam quality”, OpticsLetters vol. 25 pp. 939-941 (2000)), convert annular light beam 210 intoan anti-phase, radially polarized annular light beam 230. (Note that inthe absence of spiral phase element 232, grating 214 would convert lightbeam 210 into an in-phase, radially polarized light beam.)Alternatively, grating 214 and spiral phase element 232 are placed inthe optical path from laser 206 to mirror 212, for example between laser206 and axicon lens 216. Light beam 230 is focused onto a focal region222 by a positive axicon lens 220. Meanwhile, electron beam 204traverses apertures 226 and 228 in polarization grating 214 and lens220, respectively, to enter focal region 222. In focal region 222, thelongitudinal component of the electric field of light beam 230accelerates the electrons of electron beam 204 towards the right.

[0095] The third application is to the cutting of a workpiece. Asdescribed by V. G. Niziev and A. V. Nesterov in “Influence of beampolarization on laser cutting efficiency”, Journal of Physics D: AppliedPhysics vol. 32 pp. 1455-1461 (1999), which is incorporated by referencefor all purposes as if fully set forth herein, the laser cuttingefficiency of a radially polarized beam is 1.5 to 2 times larger thanfor plane P-polarized and circularly polarized beams. According to thepresent invention, the radially polarized beam is produced by passing alinearly or circularly polarized coherent light beam through anappropriate subwavelength dielectric grating of the present invention.

[0096] While the invention has been described with respect to a limitednumber of embodiments, it will be appreciated that many variations,modifications and other applications of the invention may be made.

What is claimed is:
 1. An optical device, for manipulating incidentlight of at most a certain maximum wavelength, comprising asubstantially planar grating formed in a dielectric material and havinga space-variant, continuous grating vector, at least a portion of saidgrating having a local period less than the maximum wavelength of theincident light.
 2. The device of claim 1, wherein said grating is formedas a plurality of grooves in a planar substrate of said dielectricmaterial.
 3. The device of claim 1, wherein a magnitude of said gratingvector varies laterally and continuously.
 4. The device of claim 1,wherein a direction of said grating vector varies laterally andcontinuously.
 5. The device of claim 1, wherein said grating vector isperiodic.
 6. The device of claim 5, wherein said grating vector istranslationally periodic.
 7. The device of claim 5, wherein said gratingvector is rotationally periodic.
 8. The device of claim 1, wherein saiddielectric material is selected from the group consisting of galliumarsenide, zinc selenide, quartz and silica glass.
 9. The device of claim1, wherein said grating is operative to pass laterally uniform,polarized incident light with a predetermined, laterally varyingtransmissivity.
 10. The device of claim 9, wherein said transmissivityvaries periodically in one lateral direction.
 11. The device of claim 1,wherein said grating is operative to pass laterally uniform, polarizedincident light with a predetermined, laterally varying retardation. 12.The device of claim 11, wherein said retardation varies periodically inone lateral direction.
 13. The device of claim 1, wherein said gratingis operative to reflect laterally uniform polarized incident light witha predetermined, laterally varying reflectivity.
 14. The device of claim13, wherein said reflectivity varies periodically in one lateraldirection.
 15. The device of claim 1, wherein said grating is operativeto reflect laterally uniform, polarized incident light with apredetermined, laterally varying retardation.
 16. The device of claim15, wherein said retardation varies periodically in one lateraldirection.
 17. The device of claim 1, wherein said grating is operativeto transform light incident thereon into a transmitted beam having apredetermined, laterally varying polarization state.
 18. The device ofclaim 17, wherein said transmitted beam has an azimuthal angle thatvaries linearly in one lateral direction.
 19. The device of claim 17,wherein said transmitted beam is radially polarized.
 20. The device ofclaim 19, wherein said radial polarization is in-phase.
 21. The deviceof claim 19, wherein said radial polarization is anti-phase.
 22. Thedevice of claim 17, wherein said transmitted beam is azimuthallypolarized.
 23. The device of claim 22, wherein said azimuthalpolarization is in-phase.
 24. The device of claim 22, wherein saidazimuthal polarization is anti-phase.
 25. The device of claim 1, whereinsaid grating is operative to transform light incident thereon into areflected beam having a predetermined, laterally varying polarizationstate.
 26. The device of claim 25, wherein said reflected beam has anazimuthal angle that varies linearly in one lateral direction.
 27. Thedevice of claim 25, wherein said transmitted beam is radially polarized.28. The device of claim 27, wherein said radial polarization isin-phase.
 29. The device of claim 27, wherein said radial polarizationis anti-phase.
 30. The device of claim 25, wherein said reflected beamis azimuthally polarized.
 31. The device of claim 30, wherein saidazimuthal polarization is in-phase.
 32. The device of claim 30, whereinsaid azimuthal polarization is anti-phase.
 33. A particle acceleratorcomprising: (a) a source of light; (b) a first optical mechanism forforming said light into an annular beam; (c) the device of claim 1, forimposing radial polarization on said annular beam: (d) a second opticalmechanism for focusing said radially polarized annular beam onto a focalregion; and (e) a particle source for directing a beam of the particleslongitudinally through said focal region.
 34. A method of cutting aworkpiece, comprising the steps of: (a) providing a beam of light; (b)imposing radial polarization on said beam of light, using the device ofclaim 1; and (c) directing said radially polarized beam at the workpieceto cut the workpiece.
 35. An apparatus for measuring a polarizationstate of light, comprising: (a) the device of claim 1; and (b) amechanism for measuring a lateral variation of an intensity of the lightafter the light has been manipulated by the device of claim
 1. 36. Amethod of modulating laterally uniform, polarized light of at most acertain maximum wavelength, comprising the steps of: (a) solving anequation ∇×{overscore (K)}(K₀,β)=0  for a grating vector {overscore (K)}that is defined by a wavenumber K₀ and by a direction β relative to areference direction, the modulation depending on β, {overscore (K)}being such that at least a portion of a grating fabricated in accordancewith {overscore (K)} has a local period less than the maximum wavelengthof the light; (b) fabricating said grating in a dielectric material inaccordance with said grating vector {overscore (K)}; and (c) directingthe light at said grating.
 37. The method of claim 36, wherein saidgrating is fabricated as a plurality of grooves in a planar substrate ofsaid dielectric material.
 38. The method of claim 36, wherein saiddielectric material is selected from the group consisting of galliumarsenide, zinc selenide, quartz and silica glass.
 39. A method ofimposing a laterally varying polarization state on light of at most acertain maximum wavelength, comprising the steps of: (a) solving anequation ∇×{overscore (K)}(K ₀,β)=0  for a grating vector {overscore(K)} that is defined by a wavenumber K₀ and by a direction β relative toa reference direction, the laterally varying polarization state being atleast partially defined by β, {overscore (K)} being such that at least aportion of a grating fabricated in accordance with {overscore (K)} has alocal period less than the maximum wavelength of the light; (b)fabricating said grating in a dielectric material in accordance with{overscore (K)}; and (c) directing the light at said grating.
 40. Themethod of claim 39, wherein said grating is fabricated as a plurality ofgrooves in a planar substrate of said dielectric material.
 41. Themethod of claim 39, wherein said reference direction is a radialdirection of a polar (r,θ) coordinate system.
 42. The method of claim39, wherein said dielectric material is selected from the groupconsisting of gallium arsenide, zinc selenide, quartz and silica glass.43. A method of measuring a polarization state of light of at most acertain maximum wavelength, comprising the steps of: (a) providing asubstantially planar grating having a transmission axis that varies inone lateral direction at least a portion of said grating having a localperiod less than the maximum wavelength of the light; (b) directing thelight at said grating; (c) measuring an intensity of the light that hastraversed said grating; and (d) determining all four Stokes parametersof the light from said intensity.
 44. The method of claim 43, furthercomprising the step of: (e) causing at least a portion of the light totraverse a polarizer subsequent to traversing said grating.
 45. Themethod of claim 43, wherein said transmission axis varies continuouslyin said one lateral direction.
 46. The method of claim 45, wherein saidtransmission axis varies linearly in said one lateral direction.
 47. Themethod of claim 43, wherein said grating is substantially planar, isformed as a plurality of grooves in a planar dielectric substrate, andhas a space-variant, continuous grating vector, said transmission axisbeing a direction of said grating vector.
 48. The method of claim 43,wherein said Stokes parameters are determined by performing respectiveintegral transforms of said intensity in said lateral direction.
 49. Amethod of measuring a polarization state of light of at most a certainmaximum wavelength, comprising the steps of: (a) providing asubstantially planar grating having a reflection axis that varies in onelateral direction at least a portion of said grating having a localperiod less than the maximum wavelength of the light; (b) directing thelight at said grating: (c) measuring an intensity of the light that isreflected from said grating; and (d) determining all four Stokesparameters of the light from said intensity.
 50. An optical device, fortransforming an incident beam of light into a transformed beam of light,comprising a substantially planar grating formed in a dielectricmaterial and having a space-variant continuous grating vector, such thatthe transformed beam is substantially free of propagating orders higherthan zero order.
 51. The device of claim 50, wherein said grating isformed as a plurality of grooves in a planar substrate of saiddielectric material.
 52. The device of claim 50, wherein a magnitude ofsaid grating vector varies laterally and continuously.
 53. The device ofclaim 50, wherein a direction of said grating vector varies laterallyand continuously.
 54. The device of claim 50, wherein said gratingvector is periodic.
 55. The device of claim 50, wherein said transformedbeam is a transmitted beam, and wherein said grating is operative topass laterally uniform, polarized incident light with a predetermined,laterally varying transmissivity.
 56. The device of claim 50, whereinsaid transformed beam is a transmitted beam, and wherein said grating isoperative to pass laterally uniform, polarized incident light with apredetermined, laterally varying retardation.
 57. The device of claim50, wherein said transformed beam is a reflected beam, and wherein saidgrating is operative to reflect laterally uniform, polarized incidentlight with a predetermined, laterally varying reflectivity.
 58. Thedevice of claim 50, wherein said transformed beam is a reflected beamand wherein said grating is operative to reflect laterally uniform,polarized incident light with a predetermined, laterally varyingretardation.
 59. The device of claim 50, wherein the transformed beam isa transmitted beam having a predetermined, laterally varyingpolarization state.
 60. The device of claim 50, wherein the transformedbeam is a reflected beam having a predetermnined, laterally varyingpolarization state.
 61. A particle accelerator comprising: (a) a sourceof light; (b) a first optical mechanism for forming said light into anannular beam; (c) the device of claim 50, for imposing radialpolarization on said annular beam: (d) a second optical mechanism forfocusing said radially polarized annular beam onto a focal region; and(e) a particle source for directing a beam of the particleslongitudinally through said focal region.
 62. A method of cutting aworkpiece, comprising the steps of: (a) providing a beam of light; (b)imposing radial polarization on said beam of light, using the device ofclaim 50; and (c) directing said radially polarized beam at theworkpiece to cut the workpiece.
 63. An apparatus for measuring apolarization state of light, comprising: (a) the device of claim 50; and(b) a mechanism for measuring a lateral variation of an intensity of thelight after the light has been transformed by the device of claim 50.64. A method of transforming an incident beam of laterally uniform,polarized light into a modulated transmitted beam, comprising the stepsof: (a) solving an equation ∇×{overscore (K)}(K₀,β)=0  for a gratingvector {overscore (K)} that is defined by a wavenumber K₀ and by adirection β relative to a reference direction, the modulation dependingon β, {overscore (K)} being such that the transmitted beam issubstantially free of propagating orders higher than zero order; (b)fabricating said grating in a dielectric material in accordance withsaid grating vector {overscore (K)}; and (c) directing the light at saidgrating.
 65. The method of claim 64, wherein said grating is fabricatedas a plurality of grooves in a planar substrate of said dielectricmaterial.
 66. A method of transforming an incident light beam into atransmitted beam upon which is imposed a laterally varying polarizationstate, comprising the steps of: (a) solving an equation ∇×{overscore(K)}(K₀,β)=0  for a grating vector {overscore (K)} that is defined by awavenumber K₀ and by a direction β relative to a reference direction,the laterally varying polarization state being at least partiallydefined by β, {overscore (K)} being such that the transmitted beam issubstantially free of propagating orders higher than zero order; (b)fabricating said grating in a dielectric material in accordance with{overscore (K)}; and (c) directing the light at said grating.
 67. Themethod of claim 66, wherein said grating is fabricated as a plurality ofgrooves in a planar substrate of said dielectric material.
 68. A methodof measuring a polarization state of an incident light beam, comprisingthe steps of: (a) providing a substantially planar grating having atransmission axis that varies in one lateral direction, said gratingbeing operative to transform the incident beam into a transmitted beamthat is substantially free of propagating orders higher than zero order;(b) directing the light at said grating; (c) measuring an intensity ofthe transmitted beam; and (d) determining all four Stokes parameters ofthe light from said intensity.
 69. A method of measuring a polarizationstate of an incident light beam, comprising the steps of: (a) providinga substantially planar grating having a reflection axis that varies inone lateral direction, said grating being operative to transform theincident beam into a reflected beam that is substantially free ofpropagating orders higher than zero order; (b) directing the light atsaid grating: (c) measuring an intensity of the reflected beam; and (d)determining all four Stokes parameters of the light from said intensity.